Transportation inequalities for stochastic differential equations with jumps

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Let \mathcal{G}_{n} denote the space of n -generated marked groups. We prove that, for every n\ge 2 , there exist 2^{\aleph_0} non-atomic, \operatorname{Out}(F_{n}) -invariant, mixing probability measures on \mathcal{G}_{n} . On the other hand, there are non-empty closed subsets of \mathcal{G}_{n} that admit no \operatorname{Out}(F_{n}) -invariant probability measure. Acylindrical hyperbolicity of the group \operatorname{Aut}(F_{n}) plays a crucial role in the proof of both results. We also discuss model-theoretic implications of the existence of \operatorname{Out}(F_{n}) -invariant, ergodic probability measures on \mathcal{G}_{n} .

Approximation of invariant measures of a class of backward Euler-Maruyama scheme for stochastic functional differential equations

For a chain transitive flow, the averages produced by the invariant probability measures are approximated by multi-valued averages of an inflated flow. The results are applied to singularly perturbed differential equations with slow and fast trajectories. Here, the multi-valued character of the averaged system is due to the presence of many invariant probability measures, often caused by resonances. Hence, the inflation of a chain transitive flow gives information on the relevance of resonances. The Takens equation modelling strong elastic constraints is treated in detail.

Let $T_{f}$ : S1 → S1 be a circle homeomorphism with two break points ab, cb that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã b, ĉb, which are the representative points ofab, cb in the interval $[0,1)$, and irrationalrotation number ρf. Suppose that $Df$ is absolutely continuous on every connected intervalof the set [0,1]\{ãb, ĉb}, that DlogDf ∈ L1([0,1]) and the product of the jump ratios of $ Df $ at thebreak points is nontrivial, i.e.$\frac{Df_{-}(\tilde{a}_{b})}{Df_{+}(\tilde{a}_{b})}\frac{Df_{-}(\tilde{c}_{b})}{Df_{+}(\tilde{c}_{b})}\ne1$.We prove, that the unique Tf - invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure onS1.

In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean--Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem.

In this paper we prove under a general condition on the relation between metric and partial order of the underlying space an equivalence to stochastic order of probability measures. Secondly we show that the set of all probability measures which are placed stochastically between two fixed probability measures is compact in the topology of weak convergence. Using weak-convergence arguments we apply these two theorems to Markov processes, getting results on bounds and existence of their invariant probability measures. Finally, we give a transience criterion for some semi-Markov processes.

Why does classical equilibrium statistical mechanics work? Malament and Zabell (1980) noticed that, for ergodic dynamical systems, the unique absolutely continuous invariant probability measure is the microcanonical. Earman and Rédei (1996) replied that systems of interest are very probably not ergodic, so that absolutely continuous invariant probability measures very distant from the microcanonical exist. In response I define the generalized properties of epsilon-ergodicity and epsilon-continuity, I review computational evidence indicating that systems of interest are epsilon-ergodic, I adapt Malament and Zabell's defense of absolute continuity to support epsilon-continuity, and I prove that, for epsilon-ergodic systems, every epsilon-continuous invariant probability measure is very close to the microcanonical.

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the BrunnMinkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality. 1. Preliminaries The setting for this paper is n-dimensional Euclidean space R. We will denote by convex figure a compact convex subset of R, and by convex body a convex figure with nonempty interior. Let Sn−1 denote the unit sphere centered at the origin o in R, and write αn−1 for the (n − 1)-dimensional volume of Sn−1. Let Bn be the closed unit ball in R, write ωn for the n-dimensional volume of Bn. Note that ωn = 2π n 2 nΓ(n2 ) , αn−1 = nωn. By a direction, we mean a unit vector, that is, an element of Sn−1. If u is a direction, we denote by u⊥ the (n − 1)-dimensional subspace orthogonal to u and by lu the line through the origin parallel to u. Denote by AGi,n the affine Grassmann manifold of i-dimensional planes in R. It is a homogeneous space under the action of the motion group G(n) (see [7], p.199). Let dξk be the normalized invariant measure of AGi,n whose restriction to the Grassmann manifold Gi,n is the invariant probability measure. Let ξ1 be a random line intersecting K. Then vol1(K ∩ ξ1) is the chord length of the intersection K ∩ ξ1. The chord power integrals of K are defined by Iλ(K) = 2αn−1 n ∫ ξ1∈AG1,n vol1(K ∩ ξ1)dξ1, 0 ≤ λ < ∞. Received November 30, 2006; Revised April 5, 2007. 2000 Mathematics Subject Classification. 52A40.

Characterization of long-term behavior of stochastic NP ecological model under regime switching

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real C∞-coboundaries of such a diffeomorphism is closed in C∞(T) if and only if its rotation number is Diophantine.

We introduce two parametrized families of piecewise affine maps on [ 0 , 1 ] 2 [0,1]^2 and [ 0 , 1 ] 3 [0,1]^3 , as generalizations of the heterochaos baker maps which were introduced and investigated by Saiki, Takahasi, and Yorke [Nonlinearity, 34 (2021), pp. 5744–5761] as minimal models of the unstable dimension variability in multidimensional dynamical systems. We show that natural coding spaces of these maps coincide with the Dyck system that has come from the theory of languages. Based on this coincidence, we start to develop a complementary analysis on their invariant probability measures. As a first attempt, we show the existence of two ergodic measures of maximal entropy for the generalized heterochaos baker maps. We also clarify a mechanism for the breakdown of entropy approachability.

Let (Φt)t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function Pt(x, ·) to π; specifically, we find conditions under which r(t)||Pt(x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.